PPF-FoldNet: Unsupervised Learning of RotationInvariant 3D Local Descriptors

Haowen Deng = C * Tolga Birdal = C Slobodan Ilic = C

= Technische Universitat Munchen, Germany C Siemens AG, Germany* National University of Defense Technology, China

Abstract. We present PPF-FoldNet for unsupervised learning of 3D local de-scriptors on pure point cloud geometry. Based on the folding-based auto-encodingof well known point pair features, PPF-FoldNet offers many desirable properties:it necessitates neither supervision, nor a sensitive local reference frame, benefitsfrom point-set sparsity, is end-to-end, fast, and can extract powerful rotation in-variant descriptors. Thanks to a novel feature visualization, its evolution can bemonitored to provide interpretable insights. Our extensive experiments demon-strate that despite having six degree-of-freedom invariance and lack of traininglabels, our network achieves state of the art results in standard benchmark datasetsand outperforms its competitors when rotations and varying point densities arepresent. PPF-FoldNet achieves 9% higher recall on standard benchmarks, 23%higher recall when rotations are introduced into the same datasets and finally, amargin of > 35% is attained when point density is significantly decreased.

Keywords: 3D deep learning, local features, descriptors, rotation invariance

1 Introduction

Local descriptors are one of the essential tools used in computer vision, easing the tasksof object detection, pose estimation, SLAM or image retrieval [23,27]. While beingwell established in the 2D domain, 3D local features are still known to lack good dis-criminative power and repeatibility. With the advent of deep learning, many areas incomputer vision shifted from hand crafted labor towards a problem specific end-to-endlearning. Local features are of course no exception. Already in 2D, learned descriptorssignificantly outperform their engineered counterparts [49,28]. Thus, it was only natu-ral for the scholars to tackle the task of 3D local feature extraction employing similarapproaches [18,51,8]. However, due to the inherent ambiguities and less informativenature of sole geometry, extracting 3D descriptors on point sets still poses an unsolvedproblem, even for learning-based methods.

Up until now, deep learning of local features in 3D has suffered from one or moreof the following: a) being supervised and requiring an abundant amount of labels inform of pairs, triplets or N -tuples [51,8], b) being sensitive to 6DoF rotations [51,8], c)involving significant hand-crafted input preparation [18] and d) unsatisfactory perfor-mance [18,30]. In this paper, we map out an elegant architecture to tackle all of theseproblems and present PPF-FoldNet: an unsupervised, high-accuracy, 6DoF transforma-tion invariant, sparse and fast 3D local feature learning network. PPF-FoldNet operates

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2 Haowen Deng, Tolga Birdal, Slobodan Ilic

mlp(64, 128, 256)

mlp(512, 512)

𝑀𝑥2𝐺𝑟𝑖𝑑

Local, orientedpoint set

4D point pair feature collection

maxpool

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Fig. 1. PPF-FoldNet: The point pair feature folding network. The point cloud local patches arefirst converted into PPF representations, and then sent into the encoder to get compressed code-words. The decoder tries to reconstruct full PPFs from these codewords by folding. This forcesthe codewords to keep the most critical and discriminative information. The learned codewordsare proven to be robust and effective as we will show across extensive evaluations.

directly on point sets, taking into account the point sparsity and permutation invariantset property, deals well with density variations, while significantly outperforming itsrotation-variant counterparts even based on the standard benchmarks.

Our network establishes theoretical rotation invariance inspired by use a point pairfeature (PPF) [4,3,8] encoding of the local 3D geometry into patches. In contrast toPPFNet [8], we do not incorporate the original points or normals into the encoding. Thecollection of these 4D PPFs are then sent to a FoldingNet-like end to end auto-encoder(AE) [48], trained to auto-reconstruct the PPFs, using a set distance. Our encoder issimpler than in FoldingNet and for decoding, we propose a similar folding scheme,where a low dimensional 2D grid lattice is folded onto a 4D PPF space and monitor thenetwork evolution by a novel lossless visualization of the PPF space. Our overall archi-tecture is based on PointNet [30] to achieve permutation invariance and to fully utilizethe sparsity. Training our AE is far easier than training, for example, 3DMatch [51],because we do not need to sample pairs or triplets from a pre-annotated large datasetand we benefit from linear time complexity to the number of patches.

Extensive evaluations demonstrate that PPF-FoldNet outperforms the state of the artacross the standard benchmarks in which severe rotations are avoided. When arbitraryrotations are introduced into the input, our descriptors outperform related approaches bya large margin including even the best competitor, Khoury et al.’s CGF [18]. Moreover,we report better performance as the input sparsifies, as well as good generalizationproperties. Our qualitative evaluations will uncover how our network operates and givevaluable interpretations. In a nutshell, our contributions can be summarized as:

– An auto-encoder, that unifies a PointNet encoding with a FoldingNet decoder,

PPF-FoldNet 3

– Use of well established 4D PPFs in this modified auto-encoder to learn rotationinvariant 3D local features without supervision.

– A novel look at the invariance of point pair features and derived from it, a new wayof visualizing PPFs and monitoring the network progress.

2 Prior Art

Following their hand-crafted counterparts [35,34,40,16,10], 3D deep learning methodsstarted to enjoy a deep-rooted history. Initial attempts to learn from 3D data used thenaive dense voxel grid representation [51,46,26,11]. While being straightforward exten-sions of 2D architectures, such networks did not perform as efficiently and robustly as2D CNNs [21]. Hence, they are superseded by networks taking into account the spatialsparsity by replacing the dense grids with octrees [33,38,43] or kd-trees [20].

Another family of works acknowledges that 3D surfaces live on 2D submanifoldsand seek to learn projections rather than the space of actual input. A reduction of di-mension to two makes it possible to benefit from developments in 2D CNNs such asRes-Nets [13]: LORAX [9] proposes a super-point to depth map projection. Kehl etal. [17] operate on the RGB-D patches that are natural projections onto the cameraplane. Huang et al. [15] anchor three local cameras to each 3D keypoint and collectmulti-channel projections to learn a semi-global representation. Cao et al. [7] use spher-ical projections to aid object classification. Tatarchenko et al. propose convolutions inthe tangent space as a way of operating on the local 2D projection [39].

Point clouds can be treated as graphs by associating edges among neighbors. Thispaves the way to the appliance of graph convolutional networks [25]. FoldingNet [48]employs graph-based encoding layers. Wang et al. [44] tackle the segmentation taskson point sets via graph convolutions networks (GCNs), while Qi et al. [32] apply GCNsto RGB-D semantic segmentation. While showing a promising direction, the currentefforts involving graphs on 3D tasks are still supervised, try to imitate CNNs and cannotreally outperform their unstructured point-processing counterparts.

Despite all developments in 3D deep learning, there are only a handful of methodsthat explicitly learn generic local descriptors on 3D data. One of the first methods thatlearns 3D feature matching, also known as correspondence, is 3DMatch [51]. It usesdense voxel grids to summarize the local geometry and learning is performed via con-trastive loss. 3DMatch is weakly supervised by task, does not learn generic descriptors,and is not invariant to rotations. PointNet [30] and PointNet++ [31] work directly on theunstructured point clouds and minimize a multi-task loss, resulting in local and globalfeatures. Similar to [51], invariance is not of concern and weak supervision is essen-tial. CGF [18] combines a hand-crafted input preparation with a deep dimensionality-reduction and still uses supervision. However, the input features are not learned but onlythe embedding. PPFNet [8] improves over all these methods by incorporating globalcontext, but still fails to achieve full invariance and expects supervision.

2.1 Background

From all of the aforementioned developments, we will now pay particular attention tothree: PointNet, FoldingNet and PPFNet which combined, give our network its name.

4 Haowen Deng, Tolga Birdal, Slobodan Ilic

PointNet [30] Direct consumption of unstructured point input in the form of a setwithin deep networks began by PointNet. Qi et al. proposed to use a point-wise multilayer perceptron (MLP) and aggregated individual feature maps into a global feature bya permutation-invariant max pooling. Irrespective of the input ordering, PointNet cangenerate per-point local descriptors as well as a global one, which can be combined tosolve different problems such as keypoint extraction, 3D segmentation or classification.While not being the most powerful network, it clearly sets out a successful architecturegiving rise to many successive studies [31,29,2,36].

FoldingNet [48] While PointNet can work with point clouds, it is still a supervised ar-chitecture, and constructing unsupervised extensions like an auto-encoder on points isnon-trivial as the upsampling step is required to interpolate sets [50,31]. Yang et al. of-fer a different perspective and instead of resorting to costly voxelizations [45], proposefolding, as a strong decoder alternative. Folding warps an underlying low-dimensionalgrid towards a desired set, specifically a 3D point cloud. Compared to other unsuper-vised methods, including GANs [45], FoldingNet achieves superior performance incommon tasks such as classification and therefore, in PPF-FoldNet we benefit fromits decoder structure, though in a slightly altered form.

PPFNet [8] proposes to learn local features informed by the global context of the scene.To do so, an N -tuple loss is designed, seeking to find correspondences jointly betweenall patches of two fragments. Features learned in this way are shown to be superior thanprior methods and PPFNet is reported to be the state-of-the-art local feature descriptor.However, even if Deng et al. stress the importance of learning permutation and rota-tion invariant features, the authors only manage to improve the resilience to Euclideanisometries slightly by concatenating PPF to the point set. Moreover, the proposed N-tuple loss still requires supervision. Our work improves on both of these aspects: It iscapable of using PPFs only and operating without supervision.

3 PPF-FoldNet

PPF-FoldNet is based on the idea of auto-encoding a rotation invariant but powerfulrepresentation of the point set (PPFs), such that the learned low dimensional embed-ding can be truly invariant. This is different to training the network with many possiblerotations of the same input and forcing the output to be a canonical reconstruction.The latter would both be approximate and much harder to learn. Input to our networkare local patches which, unlike PPFNet, are individually auto-encoded. The latent lowdimensional vector of the auto-encoder, codeword, is used as the local descriptor at-tributed to the point around which the patch is extracted.

3.1 Local Patch Representation

Our input point cloud is a set of oriented points X = xi ∈ R6, meaning that eachpoint is decorated with a local normal (e.g. tangent space) n ∈ R3: x = p,n ∈ R6.A local patch is a subset of the input Ωxr ⊂ X center around a reference point xr.

PPF-FoldNet 5

We then encode this patch as a collection of pair features, computed between a centralreference and all the other points:

FΩ = f(xr,x1) · · · f(xr,xi) · · · f(xr,xN ) ∈ R4×N−1, i 6= r (1)

The features between any pair (point pair features) are then defined to be a map f :R12 → R4 sending two oriented points to three angles and the pair distance:

f : (xTr ,x

Ti )

T → (∠(nr,d),∠(ni,d),∠(nr,ni), ‖d‖2)T (2)

d = pr − pi. An angle computation for non-normalized vectors is given in [3]. Suchencoding of the local geometry resembles that of PPFNet [8], but differs in the fact thatwe ignore the points and normals as they are dependent on the orientation and localreference frame. We instead use pure point pair features, thereby avoiding a canoni-cal frame computation. Note that the dimensionality of this feature is still irreduciblewithout data loss.Proposition 1. PPF representation f around xr explains the original oriented pointpair up to a rotation and reflection about the normal of the reference point.

Proof. Let us consider two oriented points x1 and x2. We can always write the compo-nents of the associated point pair feature f(x1,x2) as follows:

nT1 n2 = f1 nT

1 dn = f2 nT2 dn = f3 (3)

where dn = d/‖d‖. We now try to recover the original pair given its features. First, itis possible to write: nT

1

nT2

dTn

[n1 n2 dn

]=

1 f1 f2f1 1 f3f2 f3 1

(4)

given that all vectors are of unit length. In matrix notation, Eq. 4 can be written asATA = K. Then, by singular value decomposition, K = USVT and thus A =US1/2VT . Note that, any orthogonal matrix (rotation and reflection) R can now beapplied to A without changing the outcome: (RA)TRA = ATRTRA = ATA = K.Hence, such decomposition is up to finite-dimensional linear isometries: rotations andreflections. Since we know that the local patch is centered at the reference point pr = 0,we are free to choose an R such that the normal vector of pr (nr) is aligned along oneof the canonical axes, say +z = [0, 0, 1]T (freely chosen):

R = I + [v]x + [vx]2 1− nzr‖v‖

(5)

where v = nr × z, nzr is the z component of nr and I is identity. [·]x denotes skewsymmetric cross product matrix. Because now Rnr = z, any rotation θ and reflectionφ about z would result in the same vector z = Rz(θ, φ)z, ∀θ, φ ∈ R. Any pairedpoint can then be found in the canonical frame, uniquely up to two parameters as pr ←‖d‖Rz(θ.φ)Rdn, nr ← Rz(θ, φ)Rnr. utIn the case where reflections are ignored (as they are unlikely to happen in a 3D world),this leaves a single degree of freedom, rotation angle around the normal. Also note onceagain that for the given local representation, the reference point pr is common to all thepoint pairs.

6 Haowen Deng, Tolga Birdal, Slobodan Ilic

Local

patches

PPF

Signatures

Fig. 2. Visualisation of some local patches and their correspondent PPF Signatures.

Visualizing PPFs PPFs exist in a 4D space and thus it is not trivial to visualize them.While simple solutions such as PCA would work, we prefer a more geometrically mean-ingful and simpler solution. Proposition 1 allows us to compute a signature of a setof point pairs by orienting the vectors (n1,n2,d) individually for all points in orderto align the difference vectors di with the x − z plane by choosing an appropriateRz(θ.φ). Such a transformation would not alter the features as shown. In this way, thepaired points can be transformed onto a common plane (image), where the location isdetermined by the difference vector, in polar coordinates. The normal of the secondpoint would not lie in this plane but can be encoded as colors in that image. Hence, it ispossible to obtain a 2D visualization, without any data loss, i.e. all components of thevector contribute to the visualization. In Fig. 2 we provide a variety of local patch andPPF visualizations from the datasets of concern.

3.2 PPF Auto-Encoder and Folding

PPF-FoldNet employs a PointNet-like encoder with skip-links and a FoldingNet-likedecoding scheme. It is designed to operate on 4D-PPFs, as summarized in Fig. 1.

Encoder The input to our network, and thus to the encoder, is FΩ, a local PPF repre-sentation, as in §3.1. A three-layer, point-wise MLP (Multi Layer Perceptron) followsthe input layer and subsequently a max-pooling is performed to aggregate the individ-ual features into a global one, similar to PointNet [30]. The low level features are thenconcatenated with this global feature using skip-links. This results in a more powerfulrepresentation. Another two-layer MLP finally redirects these features to a final encod-ing, the codeword, which is of dimension 512.

Proposition 2. The encoder structure of PPF-FoldNet is permutation invariant.

Sketch of the proof. The encoder is composed of per-data-point functions (MLP), RELUlayers and max-pooling, all of which either do not affect the point order or are individu-ally shown to be permutation invariant [30,48]. Moreover, it is shown that compositionof functions is also invariant [48] and so is our encoder. We refer the reader to the ref-erences for further details. ut

In summary, altering the order of the PPF set will not affect the learned representation.

PPF-FoldNet 7

Decoder Our decoder tries to reconstruct the whole set of point PPFs using a singlecodeword, which in return, also forces the codeword to be informative and distill themost distinctive information from the high-dimensional input space. However, inspiredby FoldingNet, instead of trying to upsample or interpolate point sets, the decoder willtry to deform a low-dimensional grid structure guided by the codeword. Each grid pointis concatenated to a replica of the codeword, resulting in an M × 514 vector as input towhat is referred as folding operation [48]. Folding can be a highly non-linear operationand is thus performed by two consecutive MLPs: the first folding results in a deformedgrid, which is appended once again to the codewords and propagates through the secondMLP, reconstructing the input PPFs. Moreover, in contrast to FoldingNet [48], we try toreconstruct a higher dimensional set, 4D vs 3D (2D manifold); we are better off usinga deeper MLP - 5-layer as opposed to the 3-layer of [48].

Other than simplifying and strengthening the decoding, the folding is also beneficialin making the network interpretable. For instance, it is possible to monitor the gridduring subsequent iterations and envisage how the network evolves. To do so, §4.4 willtrace the PPF sets by visualizing them as described in §3.1.

Chamfer Loss Note that as size of the grid M , is not necessarily the same as the size ofthe input N , and the correspondences in 4D PPF space are lost when it comes to eval-uating the loss. This requires a distance computation between two unequal cardinalitypoint pair feature sets, which we measure via the well known Chamfer metric:

d(F, F) = max

1

|F|∑f∈F

minf∈F‖f − f‖2,

1

|F|

∑f∈F

minf∈F‖f − f‖2

(6)

where ˆ operator refers to the reconstructed (estimated) set.

Implementation details PPF-FoldNet uses Tensorflow framework [1]. The initial valuesof all variables are initialized randomly by Xavier’s algorithm. Global loss is minimizedwith an ADAM optimizer [19]. Learning rate starts at 0.001 and exponentially decaysafter every 10 epochs, truncated at 0.0001. We use batches of size 32.

4 Experimental Evaluation

4.1 Datasets and Preprocessing

To fully drive the network towards learning varieties of local 3D geometries and gainrobustness to different noises present in real data, we use the 3DMatch BenchmarkDataset [51]. This dataset is a large ensemble of the existing ones such as Analysis-by-Synthesis [41], 7-Scenes [37], SUN3D [47], RGB-D Scenes v.2 [22] and Halberand Funkhouser [12]. It contains 62 scenes in total, and we reserve 54 of them fortraining and validation. 8 are for benchmarking. 3DMatch already provided fragmentsfused from 50 consecutive depth frames of the 8 test scenes, and we follow the samepipeline to generate fragments from the training scenes. Test fragments lack the colorinformation and therefore we resort to using only the 3D shape. This also makes ournetwork insensitive to illumination changes.

8 Haowen Deng, Tolga Birdal, Slobodan Ilic

Table 1. Our results on the standard 3DMatch benchmark. Red Kitchen data is from 7-scenes [37]and the rest imported from SUN3D [47].

Spin Image [16] SHOT [35] FPFH [34] 3DMatch [51] CGF [18] PPFNet [8] FoldNet [48] Ours Ours-5K

Kitchen 0.1937 0.1779 0.3063 0.5751 0.4605 0.8972 0.5949 0.7352 0.7866Home 1 0.3974 0.3718 0.5833 0.7372 0.6154 0.5577 0.7179 0.7564 0.7628Home 2 0.3654 0.3365 0.4663 0.7067 0.5625 0.5913 0.6058 0.6250 0.6154Hotel 1 0.1814 0.2080 0.2611 0.5708 0.4469 0.5796 0.6549 0.6593 0.6814Hotel 2 0.2019 0.2212 0.3269 0.4423 0.3846 0.5769 0.4231 0.6058 0.7115Hotel 3 0.3148 0.3889 0.5000 0.6296 0.5926 0.6111 0.6111 0.8889 0.9444Study 0.0548 0.0719 0.1541 0.5616 0.4075 0.5342 0.7123 0.5753 0.6199MIT Lab 0.1039 0.1299 0.2727 0.5455 0.3506 0.6364 0.5844 0.5974 0.6234

Average 0.2267 0.2382 0.3589 0.5961 0.4776 0.6231 0.6130 0.6804 0.7182

Prior to operation, we downsample the fused fragments with spatial uniformity [5]and compute surface normals using [14] in a 17-point neighborhood. A reference pointand its neighbors within 30 cm vicinity form a local patch. The number of points in alocal patch is thus flexible, which makes it difficult to organize data into regular batches.To facilitate training as well as to increase the representation robustness to noises anddifferent point densities, each local patch is down-sampled. For a fair comparison withother methods in the literature, we use 2048 points, but also provide an extended ver-sion that uses 5K since we are not memory bound, as for example, PPFNet [8] is. Thepreparation stage ends with the PPFs calculated for the assembled local patches.

4.2 Accuracy Assessment Techniques

Let us assume that a pair of fragments P = pi ∈ R3 and Q = qi ∈ R3 arealigned by an associated rigid transformation T ∈ SE(3), resulting in a certain over-lap. We then define a non-linear feature function g(·) for mapping from input points tofeature space, and in our case, this summarizes the PPF computation and encoding as acodeword. The feature for point pi is g(pi), and g(P) is the pool of features extractedfor the points in P. To estimate the rigid transformation between P and Q, the typicalapproach finds a set of matching pairs in each fragment and associates the correspon-dences. The inter point pair set M is formed by the pairs (p,q) that lie mutually closein the feature space by applying nearest neighbor search NN :

M = pi,qi, g(pi) = NN(g(qi), g(P)), g(qi) = NN(g(pi), g(Q)) (7)

True matches set Mgnd is the set of point pairs with a Euclidean distance below athreshold τ1 under ground-truth transformation T .

Mgnd = pi,qi : (pi,qi) ∈M, ||pi −Tqi||2 < τ1 (8)

We now define an inlier ratio for M as the percentage of true matches in M as rin =|Mgnd|/|M|. To successfully estimate the rigid transformation based on M via regis-tration algorithms, rin needs to be greater than τ2. For example, in a common RANSACpipeline, achieving 99.9% confidence in the task of finding a subset with at least 3 cor-rect matches M, with an inlier ratio τ2 = 5% requires at least 55258 iterations. The-oretically, given rin > τ2, it is highly probable a reliable local registration algorithm

PPF-FoldNet 9

Table 2. Our results on the rotated 3DMatch benchmark. Red Kitchen data is from 7-scenes [37]and the rest imported from SUN3D [47].

Spin Image [16] SHOT [35] FPFH [34] 3DMatch [51] CGF [18] PPFNet [8] FoldNet [48] Ours Ours-5K

Kitchen 0.1779 0.1779 0.2905 0.0040 0.4466 0.0020 0.0178 0.7352 0.7885Home 1 0.4487 0.3526 0.5897 0.0128 0.6667 0.0000 0.0321 0.7692 0.7821Home 2 0.3413 0.3365 0.4712 0.0337 0.5288 0.0144 0.0337 0.6202 0.6442Hotel 1 0.1814 0.2168 0.3009 0.0044 0.4425 0.0044 0.0133 0.6637 0.6770Hotel 2 0.1731 0.2404 0.2981 0.0000 0.4423 0.0000 0.0096 0.6058 0.6923Hotel 3 0.3148 0.3333 0.5185 0.0096 0.6296 0.0000 0.0370 0.9259 0.9630Study 0.0582 0.0822 0.1575 0.0000 0.4178 0.0000 0.0171 0.5616 0.6267MIT Lab 0.1169 0.1299 0.2857 0.0260 0.4156 0.0000 0.0260 0.6104 0.6753

Average 0.2265 0.2337 0.3640 0.0113 0.4987 0.0026 0.0233 0.6865 0.7311

would work, regardless of the robustifier. Therefore instead of using the local regis-tration results to judge the quality of features, which would be both slow and not verystraightforward, we define M with rin > τ2 votes for a correct match of two fragments.

Each scene in the benchmark contains a set of fragments. Fragment pairs P and Qhaving an overlap above 30% under the ground-truth alignment are considered to match.Together they form the set of fragment pairs S = (P,Q) that are used in evaluations.The quality of features is measured by the recall R of fragment pairs matched in S:

R =1

|S|

|S|∑i=1

1

(rin(Si = (Pi,Qi)

)> τ2

)(9)

4.3 Results

Feature quality evaluation We first compare the performance of our features againstthe well-accepted works on the 3DMatch benchmark with τ1 = 10 cm and τ2 = 5%.Tab. 1 tabulates the findings. The methods selected for comparison comprise 3 hand-crafted features (Spin Images [16], SHOT [35], FPFH [34]) and 4 state-of-the-art deeplearning based methods (3DMatch [51], CGF [18], PPFNet [8], FoldingNet [48]). Notethat FoldingNet has never been tested on local descriptor extraction before. It is appar-ent that, overall, our PPF-FoldNet could match far more fragment pairs in comparisonto the other methods, except for scenes Kitchen and Home, where PPFNet and 3DMatchachieve a higher recall respectively. In all the other cases, PPF-FoldNet outperforms thestate of the art by a large margin,> 9% on average. PPF-FoldNet has a recall of 68.04%when using 2K sample points (the same as PPFNet), while PPFNet remains on 62.32%.Moreover, because PPF-FoldNet has no memory bottleneck, it can achieve an additonal3% improvement in comparison with the 2K version, when 5K points are used. Inter-estingly, FPFH is also constructed from a type of PPF features [34], but in a form ofmanual histogram summarization. Compared to FPFH, PPF-FoldNet has 32.15% and35.93% higher recall using 2K and 5K points respectively. It demonstrates the unprece-dented strength of our advanced method in compressing the PPFs. In order to optimallyreconstruct PPFs in the decoder, the network forces the bottleneck codeword to be com-pact as well as distilling the most critical and distinctive information in PPFs.

10 Haowen Deng, Tolga Birdal, Slobodan Ilic

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Fig. 3. Evaluations on 3DMatch benchmark: (a) Results of different methods under varying in-lier ratio threshold (b) Results of different methods under varying point distance threshold (c)Evaluating robustness again point density (d) Evaluations against rotations around z-axis

To illustrate that parameters in the evaluation metric are not tuned for our own good,we also repeat the experiments with different τ1 and τ2 values. The results are shownin Fig. 3(a) and Fig. 3(b). In Fig. 3(a), τ1 is fixed at 10 cm, τ2 increases gradually from1% to 20%. When τ2 is above 4%, PPF-FoldNet always has a higher recall than theother methods. Below 4%, some other methods may obtain a higher recall but this istoo strict for most of the registration algorithms anyway. It is further noteworthy thatwhen τ2 is set to 20%, the point where PPF-FoldNet still gets a recall above 20%, theperformance of the other methods falls below 5%. This justifies that PPF-FoldNet iscapable of generating many more sets of matching points with a high inlier ratio rin.This offers a tremendous benefit for the registration algorithms. In Fig. 3(b), τ2 is fixedat 5%, τ1 increases gradually from 0 cm to 20 cm. When τ1 is smaller than 12 cm, PPF-FoldNet consistently generates higher recall. This finding indicates that PPF-FoldNetmatches more point pairs with a small distance error in the Euclidean space, whichcould efficiently decrease the rigid transformation estimation errors.

Tests on rotation invariance To demonstrate the outstanding rotation-invariance prop-erty of PPF-FoldNet, we take random fragments out of the evaluation set and graduallyrotate them around the z-axis from 60 to 360 in steps of 60. The matching resultsare shown in Fig. 3(d). As expected, both PPFNet and 3DMatch perform poorly as theyoperate on rotation-variant input representations. Hand crafted features or CGF alsodemonstrate robustness to rotations thanks to the reliance on the local reference frame(LRF). However, PPF-FoldNet stands out as the best approach with a much higher recallwhich furthermore does not require computation of local reference frames.

To further test how those methods perform under situations with severe rotations,we rotate all the fragments in 3DMatch benchmark with randomly sampled axes and an-gles over the whole rotation space, and introduce a new benchmark – Rotated 3DMatchBenchmark. The same evaluation is once again conducted on this new benchmark.Keeping the accuracy evaluations identical, our results are shown in Tab. 2. 3DMatchand PPFNet completely failed under this new benchmark because of the variables in-troduced by large rotations. Once again, PPF-FoldNet, surpasses all other methods,achieving the best results in all the scenes, predominates the runner-up CGF by largemargins of 18.78% and 23.24% respectively when using 2K and 5K points.

PPF-FoldNet 11

Table 3. Accuracy comparison of different PPF representations.

Kitchen Home 1 Home 2 Hotel 1 Hotel 2 Hotel 3 Study MIT Lab Average

PPFH 0.534 0.622 0.486 0.341 0.346 0.574 0.233 0.351 0.436Bobkov1 0.514 0.635 0.510 0.403 0.433 0.611 0.281 0.481 0.483Our-PPF 0.506 0.635 0.495 0.350 0.385 0.667 0.267 0.403 0.463

Sparsity evaluation Thanks to the sparse representation of our input, PPF-FoldNet isalso robust in respect of the changes in point cloud density and noise. Fig.3(c) showsthe performance of different methods when we gradually decrease the points in thefragment from 100% to only 6.25%. We can see that PPF-FoldNet is least affected bythe decrease in point cloud density. In particular, when only 6.25% points are left in thefragments, the recall for PPF-FoldNet is still greater than 50% while PPFNet remainsaround 12% and the other methods almost fail. The results of PPFNet and PPF-FoldNettogether demonstrate that PPF representation offers more robustness in respect of pointdensities, which is a common problem existing in many point cloud representations.

Can PPF-FoldNet operate with different PPF constructions? We now study 3 identi-cal networks, trained for 3 different PPF formulations: ours, PPFH (the PPF used inFPFH [34]) and Bobkov1 et al. [6]. The latter has an added component of occupancyratio based on grid space. We use a subset of 3DMatch benchmark to train all networksfor a fixed number of iterations and test on the rotated fragments. Tab. 3 presents ourfindings: all features perform similarly. Thus, we do not claim the superiority of ourPPF representation, but stress that it is simple, easy to compute, intuitive and easy tovisualize. Due to the voxelization, Bobkov1 is significantly slower than the other meth-ods, and due to the lack of an LRF, our PPF is faster than PPFH’s. Using stronger pairprimitives would favor PPF-FoldNet as our network is agnostic to the PPF construction.

Runtime We run our algorithm on a machine loaded with NVIDIA TitanX Pascal GPUand an Intel Core i7 3.2GHz CPU. On this hardware, computing features of an entirefragment via FPFH [34] takes 31.678 seconds, whereas PPF-FoldNet achieves a 10×speed-up with 3.969 seconds, despite having similar theoretical complexity. In particu-lar, our input preparation for PPF extraction runs in 2.616 seconds, and the inference in1.353. This is due to 1) PPF-FoldNet requiring only a single pass over the input, 2) ourefficient network accelerated on GPU powered Tensorflow.

4.4 Qualitative Evaluations

Visualizing the matching result From the quantitative results, PPF-FoldNet is expectedto have better and more correct feature matches, especially when arbitrary rigid trans-formations are applied. To show this visually, we run different methods and ours acrossseveral fragments undergoing varying rotations. In Fig. 4 we show the matching regions,over uniformly sampled [5] keypoints on these fragments. It is clear that our algorithmperforms the best among all others in discovering the most correct correspondences.

12 Haowen Deng, Tolga Birdal, Slobodan Ilic

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Fig. 4. Qualitative results of matching across different fragments and for different methods. Whensevere transformations are present, only hand-crafted algorithms, CGF and our method achievessatisfactory matches. However, for PPF-FoldNet, the number of matches are significantly larger.

Monitoring network evolution As our network is interpretable, it is tempting to qualita-tively analyze the progress of the network. To do that we record the PPF reconstructionoutput at discrete time steps and visualize the PPFs as explained in § 3.1. Fig 5 showssuch a visualization for different local patches. First, thanks to the representation power,our network achieves high fidelity recovery of PPFs. Note that even though the network

PPF-FoldNet 13

Local Patch Original PPF 𝑇 = 1 𝑇 = 4 𝑇 = 7 𝑇 = 10 𝑇 = 70

Fig. 5. Visualizing signatures of reconstructed PPFs. As the training converges, the reconstructedPPF signatures become closer to the original signatures. Our network reveals the underlyingstructure of the PPF space.

starts from a random initialization, it can quickly recover a desired point pair featureset, even after only a small number of iterations. Next, for similar local patches (top andbottom rows), the reconstructions are similar, while for different ones, different.

Visualizing the latent space We now attempt to visualize the learned latent space andassess whether the embedding is semantically meaningful. To do so, we compute a setof codewords and the associated PPF signatures. We then run the Barnes Hut T-SNEalgorithm [24,42] on the extracted codewords and form a two-dimensional embeddingspace, as shown in Fig. 6. At each 2D location we paint the PPF signature and therebyillustrate the distribution of PPFs along the manifold. We also plot the original patcheswhich generated the codewords and their corresponding signatures as cutouts. Presentedin Fig. 6, whenever the patches are geometrically and semantically close, the computeddescriptors are close, and whenever the patches have less physical similarity, they areembedded into different parts of the space. This provides insight into the good perfor-mance and meaningfulness in the relationships our network could learn. In a furtherexperiment, we extract a feature at each location of the point cloud. Then, we reducethe dimension of the latent space to three via TSNE [24], and colorize each point by thereduced feature vector. Qualitatively justifying the repeatibility of our descriptors, theoutcome is shown in Fig. 7. Note that, descriptors extracted by the proposed approachlead to similar colors in matching regions among the different fragments.

5 Concluding Remarks

We have presented PPF-FoldNet, an unsupervised, rotation invariant, low complexity,intuitive and interpretable network in order to learn 3D local features solely from point

14 Haowen Deng, Tolga Birdal, Slobodan Ilic

Fig. 6. Visualization of the latent space of codewords, associated PPFs and samples of clusteredlocal 3D patches using TSNE [24,42].

Fig. 7. Visualization of the latent feature space on fragments fused from different views. To mapeach feature to a color on the fragment, we use TSNE embedding [24]. We reduce the dimensionto three and associate each low dimensional vector with an RGB color.

geometry information. Our network is built upon its contemporary ancestors, PointNet,FoldingNet & PPFNet and it inherits best attributes of all. Despite being rotation in-variant, we have outperformed all the state-of-the-art descriptors, including supervisedones even in the standard benchmarks under challenging conditions with varying pointdensity. We believe PPF-FoldNet offers a promising new approach to the importantproblem of unsupervised 3D local feature extraction and see this as an important steptowards unsupervised revolution in 3D vision.

Our architecture can be extended in many directions. One of the most promising ofthose would be to adapt our features towards tasks like classification and object poseestimation. We conclude with the hypothesis that the generalizability in our unsuper-vised network should transfer easily into solving other similar problems, giving rise toan open application domain.

PPF-FoldNet 15

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A Appendix

A.1 Evaluations on Generalizability

Unfortunately, state-of-the-art methods for learning 3D features rely heavily on theavailability of extensively annotated data - such as ground truth matches between thepairs. In 3DMatch, CGF and PPFNet, contrastive, triplet and N-tuple losses are usedrespectively. Such supervision prevents these methods from immediately extending todifferent datasets, without fine-tuning. It might occasionally be prohibitive to even ob-tain labeled data for new datasets.

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This is different for PPF-FoldNet,where we learn a completely unsuper-vised representation. Thanks to the novelauto-encoder, we can operate on anyavailable dataset without requiring auxil-iary label information. Therefore, we aremotivated to believe that PPF-FoldNetwould generalize better to unseen data,even with a small subset of unlabeleddata being available.

To test the aforementioned hypoth-esis, we propose an experimentation,where PPF-FoldNet is trained on a smallportion of scenes and tested on multipledifferent ones. We divide the Chess scenefrom 7-scenes dataset into train and testsplits, train PPF-FoldNet from scratch, and measure the loss on the test data includingChess and other six scenes as well. Note that the remaining six scenes do not con-tribute to training. For all datasets, we plot the loss curves among iterations in Fig. 8.The dashed line stands for average loss of training data for each epoch, and the othercurves depict the average loss of test data from each scene respectively. As the trainingproceeds, the losses for all 7 datasets decrease following similar trend. Achieving suchresidual values validates that PPF-FoldNet can generalize to unseen input.

A.2 Additional Visualizations of Matching

Fig. 9 presents further qualitative analysis involving matching of rotated fragments andacross all evaluated methods.

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Fig. 9. Qualitative results of matching across different fragments and for different methods.When transformations involving rotations are present, only hand-crafted algorithms, CGF andour method achieves satisfactory matches. However, for our PPF-FoldNet, the number of matchesare significantly larger.